## 1. Introduction

[2] Quasi-planar structures (QPS) play an important role in many electromagnetic (EM) applications, including radiation and scattering from rough surfaces, microstrip structures, microwave integrated circuits, and optical gratings. Several efficient numerical methods have been proposed recently [*Tsang et al.*, 1995; *Johnson*, 1998; *Chou and Johnson*, 1998, 2000; *Jandhyala et al.*, 1998a, 1998b; *Bindiganavale et al.*, 1998; *Torrungrueng et al.*, 2000; *Torrungrueng and Johnson*, 2001a, 2001b, 2001c; *Hu and Chew*, 2000; *Valero-Nogueira and Rojas*, 2000] to obtain an accurate description of EM scattered fields for electromagnetically large structures. One of these methods, the novel spectral acceleration (NSA) algorithm, has been shown to be an extremely efficient (*N*_{tot}) iterative method of moments (MOM) for the computation of scattering from both one-dimensional (1-D) and two-dimensional (2-D) large-scale QPS [*Chou and Johnson*, 1998; *Torrungrueng et al.*, 2000; *Torrungrueng and Johnson*, 2001a, 2001b, 2001c; *Chou and Johnson*, 2000; *Valero-Nogueira and Rojas*, 2000]. This method accelerates the matrix-vector multiplication in an iterative MOM solution and divides contributions between points into “strong” (exact matrix elements) and “weak” (NSA algorithm) regions. The NSA method is based on a spectral representation of the electromagnetic Green's function and appropriate contour deformation, resulting in a fast multipole-like formulation in which contributions from large numbers of points to a single point are evaluated simultaneously. Unlike traditional multipole methods, however, only one large group of points is considered for the calculation of weak region contributions, resulting in a more efficient computation. Like the steepest descent fast multipole method [*Jandhyala et al.*, 1998a, 1998b], to further improve efficiency of the spectral expansion, the angular spectral integration path is deformed in the complex angular plane, resulting in a smaller domain of integration with less rapid oscillation along the path. The multiplication is performed in a forward sweep followed by a backward sweep, with the weak region continuously increasing in size as the multiplication proceeds in one direction. Because of the use of forward and backward sweeps, the NSA approach is well suited for incorporation into the “forward-backward” (FB) iterative method [*Kapp and Brown*, 1996; *Holliday et al.*, 1996; *Toporkov et al.*, 1998] but can also be used in any standard iterative method. Details of the NSA algorithm are provided by *Chou and Johnson* [1998].

[3] In the NSA method the most important issue is to determine the appropriate NSA parameters, which include the tilt angle (δ) of the deformed contour in the complex angular domain, the domain of integration ([−ϕ_{max}, ϕ_{max}]), and the integration step size (Δϕ). In the original paper [*Chou and Johnson*, 1998] these parameters are derived on the basis of the assumption that the outermost possible saddle point, ϕ_{s,max}, along the real axis in the complex angular (ϕ) plane is small. For a given surface height variation this assumption can be satisfied by adjusting the size of the strong region. However, for QPS with large height variations the adjusted size of the strong region is typically large, resulting in significant increases in computational time for the strong-region contribution and degrading overall efficiency of the NSA algorithm. In addition, for the case of 1-D extremely large scale structures, studies based on the physical optics (PO) approximation and a flat surface assumption show that the given 1-D NSA parameters of *Chou and Johnson* [1998]may yield inaccurate results. Inaccuracy comes from the fact that the complex radiation function (plane wave spectrum) of a source group far separated from the receiving element is rapidly decayed along the deformed contour away from the origin in the ϕ plane, requiring a higher sampling rate in ϕ to retain accuracy. Analytical results obtained from asymptotic evaluations of the radiation integral associated with the PO and flat surface assumptions suggest that the very large weak region associated with 1-D extremely large scale QPS should be decomposed into more than one separate weak region, and appropriate NSA parameters must be determined separately for each weak region to improve accuracy. Thus the new proposed scheme for the NSA algorithm can be classified as a “multilevel” algorithm. Note that the multilevel NSA algorithm is distinct from standard multilevel algorithms applied in other methods [*Brandt*, 1991; *Lu and Chew*, 1994; *Song and Chew*, 1995].

[4] This paper is organized as follows: Section 2 presents the derivation of analytical formulas associated with the 1-D NSA parameters for an arbitrary value of ϕ_{s,max}. The “multilevel” NSA algorithm for 1-D extremely large scale QPS is illustrated in section 3. Some numerical results are presented in section 4, and a summary and conclusions can be found in section 5. An *e*^{−iωt} time harmonic convention is assumed and suppressed throughout this paper, and the propagation constant is defined as where ω is the radian frequency and ϵ and μ are the permittivity and permeability of free space, respectively.